Syllabus

Measurements in Physics

Chapter 1

1.1 Distance, Time, and Mass Measurements

Physics experiments involve the measurement of a variety of quantities.

These measurements should be accurate and reproducible.

The first step in ensuring accuracy and reproducibility is defining the units in which the measurements are made.

1.1 Distance, Time, and Mass Measurements

OLD 1 meter

1 meter, distance traveled by light (vacuum), in 1/299,792,458 of a second

1 second 1 kg

SI units meter (m): unit of length

kilogram (kg): unit of mass

second (s): unit of time

1.1 Distance, Time, and Mass Measurements

The units for length, mass, and time (as well as a few others), are regarded as base SI units.

These units are used in combination to define additional units for other important physical quantities such as force and energy.

1.1 Distance, Time, and Mass Measurements

1.1 Distance, Time, and Mass Measurements

1.2 Converting Units

THE CONVERSION OF UNITS

1 ft = 0.3048 m

1 mi = 1.609 km

1 hp = 746 W

1 liter = 10-3 m3

The World’s Highest Waterfall

The highest waterfall in the world is Angel Falls in Venezuela, with a total drop of 979.0 m. Express this drop in feet.

What if you can’t remember there are 3.281 feet in a meter? What do you remember? Perhaps 1 inch = 2.54 cm (yes?)

Also, 12 inches = 1 foot, and 100 cm = 1 m.

Since 3.281 feet = 1 meter, it follows that 1m = (1 m) 100 cm

m⎛⎝⎜

⎞⎠⎟

12.54

inchcm

⎛⎝⎜

⎞⎠⎟

112

ftinch

⎛⎝⎜

⎞⎠⎟= (1 m) 100

(2.54)(12)⎛⎝⎜

⎞⎠⎟

ftm

⎛⎝⎜

⎞⎠⎟= 3.281ft

979.0 meters = 979.0 meters( ) 3.281 feet

1 meter⎛⎝⎜

⎞⎠⎟= 3212 feet

= 1

1.2 Converting Units

= 1 = 1

= 1

Reasoning Strategy: Converting Between Units

1.  In all calculations, write down the units explicitly.

2.  Treat all units as algebraic quantities. When identical units are divided, they are eliminated algebraically.

3.  Use the conversion factors located on back of front and rear covers. Be guided by the fact that multiplying or dividing an equation by a factor of 1 does not alter the equation.

1.2 Converting Units

DIMENSIONAL ANALYSIS [L] = length [M] = mass [T] = time

x = 12 v t2

Is the following equation dimensionally correct?

L⎡⎣ ⎤⎦ =

LT

⎡

⎣⎢

⎤

⎦⎥ T⎡⎣ ⎤⎦

2= L⎡⎣ ⎤⎦ T⎡⎣ ⎤⎦

1.3 Fundamental Constants and Dimensional Analysis

A speed is a length divided by a time

Speed v, units are

L⎡⎣ ⎤⎦T⎡⎣ ⎤⎦

Examples meters/second miles/hour

1.4 Measurement, Uncertainty, and Significant Figures

Measure the radius and height of this cylinder. Using a “1 meter ruler”, smallest division is 1 mm. (Need to make sure ruler is “accurate”)

Measure radius – but it is hard to find the center. Due to sharp edges, meaurement of diameter is more “precise”. Measure diameter and divide by 2.

Each dimension of the cylinder has 3 “significant figures”

Any function of these dimensions also has 3 significant digits, no more – no less! But don’t round off intermediate calculations.

Measurements: d = 272 ±1 mm h = 409 ±1 mmCalculation: r = 136 ±1 mm

Dimensions: r = 13.6 cm h = 40.9 cm

wood and metal rulers agree at only 1 temperature

1.4 Measurement, Uncertainty, and Significant Figures

What is the volume of the cylinder in m3 ?

Volume = [Area of circle](Height) Using a calculator gives:

Volume is NOT known more “precisely” than input dimensions.

Dimensions: r = 13.6 cm h = 40.9 cm

V = πr 2⎡⎣ ⎤⎦(h) = (3.14159...)(13.6cm)2⎡⎣ ⎤⎦(40.9cm) = 23,765.72... cm3

>>3 significant figures??

V = 2.38×104cm3 (23,800 cm3)Not done yet! Need answer in m3 !

V = (2.38×104)(cm3) 1m100cm

⎛⎝⎜

⎞⎠⎟

3

= 2.38×10−2 m3

Kinematics in One Dimension Kinematics describes motion,

and Dynamics investigates what causes motion

Chapter 2

Speed v : can only be positive

Velocity vx : value with sign indicating direction (in x)

⎫⎬⎪

⎭⎪

Instantaneousat the time t

Absolute value of the velocity is the speed : v = vx

Example : Choose "to the right" as positive. Object's speed is v = 20 m/s. If object is moving to the right, the velocity, vx = 20 m/s. If object is moving to the left, the velocity, vx = −20 m/s.

In Chapter 2: All motion is along a 1D line and is called the x-axis. YOU decide which direction along x is POSITIVE.1D line can be Horizontal, for motion of a car, boat, or human. 1D line can be Vertical, for objects dropped or thrown upward.ID line can be a Diagonal, for objects moving on a ramp.

2.1 Motion in one dimension (definitions)

vx = +v vx = +vSliding block

Thrown upward

vx = −v

vx = −vOR

Direction choice Direction choice

Later motion when falling downward

OR

If you determine that vx = −20m/s it must be moving to the left.

If you determine that vx = −20m/s it must be moving downward.

2.1 Motion in one dimension (examples)

Horizontal motion Vertical motion Speed v : is always positive

Moving: How can one tell if an object is moving at time, t ?

Look "at a few times a little bit" earlier. Then look "at a few times a little bit" later. Check if the object is at the same place as it was at time t! If the position of the object changes --- it is MOVING at time t ! If the position of the object doesn't change, it is NOT MOVING at time t!

(stationary) Example :

If an object is thrown upward, at the one time it reaches its highest point its speed v is 0 (instantaneously) but the object IS MOVING at that time! Turning around to a new direction is motion. It is MOVING.

Zero speed at one time t is NOT EQUIVALENT to not moving.

2.1 Motion in one dimension (definitions)

2.1 Motion in one dimension (Displacement and Distance)

Δx = x − x0 = displacement

x0 x +x

Δx Displacement = Δx

x0 = position (at t = 0) x = final position

Since x > x0 , then displacement Δx is positive

t t0

Origin

The travel distance d = Δx is always positive.

Direction choice

2.1 Motion in one dimension (Displacement and Distance)

Origin

t

Displacement = Δx +x

x0 x

x = final position

Δx = x − x0 = displacement

t0

If x0 > x, then displacement Δx will be negative

The travel distance, d = Δx is always positive.

x0 = initial position (at t = 0)

Direction choice

2.2 Speed and Velocity

Average speed is the distance traveled divided by the time (t - t0) required to cover the distance.

Average speed, v = Distance

Elapsed timeSI units for speed: meters per second (m/s)

ClearlyDistance = Average speed( ) Elapsed time( ) d = vΔt

Average value of v, is written as v (v with a bar over it).

Clicker Question 2.1

A jogger travels 1500 m at an average speed of 2.00 m/s. How long did it take to cover the distance ?

a) 300 seconds b) 750 seconds c) 3000 seconds d) 1700 seconds e) 10 minutes

Clicker Question 2.1

A jogger travels 1500 m at an average speed of 2.00 m/s. How long did it take to cover the distance ?

a) 300 seconds b) 750 seconds c) 3000 seconds d) 1700 seconds e) 10 minutes

Average speed

v = d (distance)Δt (elapsed time)

Δt = dv= 1500 m

2.00 m/s= 750 s

2.2 Speed and Velocity

Average velocity is the displacement divided by the elapsed time.

vx =

x − x0

t − t0= ΔxΔt

This average places no restriction on how the velocity has changed over time. For example, it could reverse direction a number of times over the time of the displacement.

2.2 Speed and Velocity

Example: The World’s Fastest Jet-Engine Car

Andy Green in the car ThrustSSC set a world record of 341.1 m/s in 1997. Two runs are made through the course, one in each direction. From the data shown, determine the average velocity for each run.

Δx

Δx

vx =

ΔxΔt

= +1609 m4.740 s

= +339.5m s

Average velocity run 1

vx =

ΔxΔt

= −1609 m4.695 s

= −342.7m s

Average velocity run 2

negative Average speed

v = v1 + v2

2= 341.1 m/s

2.2 Speed and Velocity

The instantaneous velocity indicates how fast the car moves and the direction of motion at each instant of time.

vx = lim

Δt→0

ΔxΔt